An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once.

Adding this to the known dynamics of each body segment, enables the solution of equations based on the Newton – Euler equations of motion permitting computations of the net forces and the net moments of force about each joint at every stage of the gait cycle.

* If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the line graph of G is Hamiltonian.

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus.

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere.

Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.

It states that the total Gaussian curvature of such a closed surface is equal to 2π times the Euler characteristic of the surface.

It holds true however for the compact closed unit disc, which also has Euler characteristic 1, because of the added boundary integral with value 2π.

For closed surfaces, this classification is consistent with the Gauss – Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic.

* the constant coefficient a < sub > 0 </ sub > is the Euler characteristic of P. When P is a closed convex polytope, a < sub > 0 </ sub > = 1.

In mathematics, the generalized Gauss – Bonnet theorem ( also called Chern – Gauss – Bonnet theorem ) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature.

* Leonhard Euler uses closed curves ( which become known as Euler diagrams ) to illustrate syllogistic reasoning.

The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic.

where is the mean curvature, is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss – Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic of the surface, so

Next, Euler observed that ( except at the endpoints of the walk ), whenever one enters a vertex by a bridge, one leaves the vertex by a bridge.

In modern language, Euler shows that the possibility of a walk through a graph, traversing each edge exactly once, depends on the degrees of the nodes.

Such a walk is now called an Eulerian path ( oy • lɛr • i • ən ) or Euler walk in his honor.

Such a walk is called an Eulerian circuit or an Euler tour.

As in the classic problem, no Euler walk is possible ; coloring does not affect this.

An Eulerian trail, or Euler walk in an undirected graph is a path that uses each edge exactly once.

To create RSA signature keys, generate an RSA key pair containing a modulus N that is the product of two large primes, along with integers e and d such that e d ≡ 1 ( mod φ ( N )), where φ is the Euler phi-function.

In mathematics, a Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair ( s, t ), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.

For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler.

* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).

Solovay and Strassen showed that for every composite n, for at least n / 2 bases less than n, n is not an Euler – Jacobi pseudoprime.

It can be shown that for any odd composite n, at least ¾ of the bases a are witnesses for the compositeness of n. The Miller – Rabin test is strictly stronger than the Solovay – Strassen primality test in the sense that for every composite n, the set of strong liars for n is a subset of the set of Euler liars for n, and for many n, the subset is proper.

Another particularly useful way to represent the Euler line is in terms of a parameter t. Starting with the circumcenter ( with trilinears ) and the orthocenter ( with trilinears, every point on the Euler line, except the orthocenter, is given as

Based on, we have:, where is the Euler – Mascheroni constant or, more generally, for every n we have:

It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.

An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.

In formal language, gimbal lock occurs because the map from Euler angles to rotations ( topologically, from the 3-torus T < sup > 3 </ sup > to the real projective space RP < sup > 3 </ sup >) is not a covering map – it is not a local homeomorphism at every point, and thus at some points the rank ( degrees of freedom ) must drop below 3, at which point gimbal lock occurs.

Euler angles provide a means for giving a numerical description of any rotation in three dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space ( rotations ) can be realized by a change in the source space ( Euler angles ).

It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the first paper on graph theory, that every cubic graph has an even number of vertices.

He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal Latin squares of order 4n + 2 for every n.

In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex.

* Half loop ( also known as a " Euler " or " Thoren "), a full-rotation jump with a loop entry but landed on the back inside edge of the opposite foot.

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

Euler ’ s Disk has an optimized aspect ratio and a precision polished, slightly rounded edge to maximize the spinning / rolling time.

A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.